Diffusive limit of a kinetic model for cometary flows
type: Item
instance of: scholarly article
title: Diffusive limit of a kinetic model for cometary flows
zbMATH Open document ID: 1181.82055
author: Christian Schmeiser
author: Jan Haškovec
published in: Journal of Statistical Physics
publication date: 2009-09-09T00:00:00Z
review text: A kinetic equation with a relaxation time model for wave-particle collisions is considered, motivated by the analysis of cometary flows and cosmic-ray transport in astrophysical plasmas [\textit{L. L. Williams} and \textit{J. R. Jokipii}, Astrophys. J. 417, 725--734 (1993)]. Similarly to the BGK-model of gas dynamics, it involves a projection onto the set of equilibrium distributions isotropic around the velocity. This equation depends nonlinearly on the moments of the distribution function. In the two first sections of the paper, the authors provide a short and useful presentation of the relevant properties of this equation and on the mathematical results obtained about it since its proposal in 1996. In Section 3, the macroscopic limit is formally computed using diffusive and low Mach number scaling. The result is a generalization of the incompressible Navier-Stokes equations, where the momentum equations are coupled to a diffusive equation for an energy distribution function. In Section 4, an entropy inequality and some other properties of the limiting system are discussed, and a simplified version is obtained by a moment approximation around a local Maxwellian energy distribution. The resulting system can be related to a low Mach number model with constant viscosity of fluid mechanics, which already appeared in the literature [\textit{P. Embid}, Commun. Partial Differ. Equ. 12, 1227--1283 (1987; Zbl 0632.76075)]. The authors discuss the difficulties that hinder to extend to general energy distributions the weak existence theory known for this equation [\textit{P.-L. Lions}, Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)]. Finally, in Section 5, for a linearized version around global equilibrium corresponding to Stokes flow for the divergence-free set of the velocity and a decoupled nonlocal equation for the energy distribution, an existence result for initial value problems is proved by using th…
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 82D05
zbMATH DE Number: 5602584
zbMATH Keywords: kinetic equation
zbMATH Keywords: wave-particle collision operator
zbMATH Keywords: cometary flows
zbMATH Keywords: diffusive scaling
zbMATH Keywords: macroscopic limit
zbMATH Keywords: entropy inequality
Wikidata QID: Q62568233
reviewed by: D. Jou
MaRDI profile type: Publication
full work available at URL: s10955-009-9784-3
OpenAlex ID: W1991615825
cites work: The mathematical theory of dilute gases
cites work: On the viscosity and thermal conduction of fluids with multivalued internal energy
cites work: Turbulence models for incompressible fluids derived from kinetic theory
cites work: Q4896493
cites work: On the macroscopic dynamics induced by a model wave-particle collision operator
cites work: Existence of solutions of a kinetic equation modeling cometary flows
cites work: A kinetic description of anisotropic fluids with multivalued internal energy.
cites work: Nonlinear diffusion as limit of kinetic equations with relaxation collision kernels
cites work: Well-posedness of the nonlinear equations for zero mach number combustion
cites work: Q4399897
cites work: Classification of equilibrium solutions of the cometary flow equation and explicit solutions of the Euler equations for monatomic ideal gases
cites work: Existence and convergence to equilibrium of a kinetic model for cometary flows
cites work: Convergence to Equilibrium for the Linearized Cometary Flow Equation
cites work: Q4389323
cites work: Compressible fluid flow and systems of conservation laws in several space variables
cites work: Q4256067
DOI: 10.1007/S10955-009-9784-3
description: scientific article; zbMATH DE number 5602584